Temperature effects on the calculation of the functional derivative of Tc with respect to α2F(ω)

The functional derivative of the superconducting transition temperature Tc with respect to the electron-phonon coupling function α2F(ω),δTc2/δα2F(ω) permits identifying the frequency regions where phonons are most effective in raising Tc. This work presents an analysis of temperature effects on the calculation of the δTc/δα2F(ω) and μ* parameters. The results may permit establishing that the variation of the temperature in the δTc/δα2F(ω) and μ* parameter allows establishing patterns and conditions that are possibly related to the physical conditions in the superconducting state, with implications on the theoretical estimation of the Tc.

The authors wish to acknowledge all the suggesDons and comments given to this work by the reviewers. The ideas proposed by reviewers are very interesDng and consistent with our work. The correcDons made considerably improved the manuscript.
We present point-by-point responses to the comments from reviewers:

For Reviewer 1:
(1) I cannot agree with the sentence: "The funcDon α2F(ω) is obtained from the soluDon of the Eliashberg equaDons" (page 3, line 62). It is generally known that the funcDon α2F(ω) is an input parameter (together with the Coulomb pseudopotenDal) to the Eliashberg equaDons. Eliashberg spectral funcDon can be obtained theoreDcally (DFT calculaDons) or experimentally (tunneling experiment).

Answer:
We recognize the error in the men.oned sentence. So, the last part of the paragraph that contains the men.oned sentence was adjusted as follows: "All that informaDon is gathered in a funcDon, the Eliashberg spectral funcDon, or electron-phonon coupling funcDon α2F(ω) (see Fig.1a), which can be obtained both theoreDcally (DFT calculaDons) and experimentally (tunneling experiment). The Eliashberg spectral funcDon is obtained from the calculated phonon spectrum and the calculated electron-phonon matrix elements [cites]. The Coulombic repulsion between electrons was included through a parameter μ" As can be seen in the manuscript: (2) Paper is very interesDng but more technical informaDon is needed. Did Authors use their codes and are they widely available? It would be great to can verify their correct operaDon. Here, the funcDonal derivaDves were obtained with the procedure widely used by Carbo_e et al.
[cites] which is based on the work of G. Bergmann and D. Rainer [cite].

As can be seen in the manuscript:
(3) It is possible to calculate the funcDonal derivaDve of Tc on the base of the Allen-Dynes equaDon? Tc calculated using the Allen-Dynes equaDon depends on the a2F funcDon.
Answer: The cri.cal temperature formula proposed by Allen-Dynes could be understood as a result of the adjustment of the Eliashberg model with a tendency to the BCS limit. The Allen-Dynes equa.on depends indirectly on the func.on α 2 F(ω) through the electron-phonon coupling parameter (λ). This implies the possibly that δTc/δα 2 F(ω) could be obtained from it. We infer that this possible procedure would not achieve the generality of the one proposed by G. This subject is interesDng, but we think that it must be analyzed in greater detail before making any proposal. For this reason, this idea will not be included in this manuscript.

For Reviewer 2:
(1) It would be helpful to provide more context about the significance of the results for nonexperts in the field. While the authors briefly menDon the importance of the behavior of δTc/δα2 F(ω) for determining opDmal physical condiDons of the superconducDng state and esDmaDng Tc, more informaDon could be provided about why this is important and how it relates to current research in the field.

Answer: The following contextualiza.on paragraph is included in the manuscript (introduc.on sec.on)
"One of the possible contribuDons of theoreDcal physics in superconducDvity is to clearly establish the fundamental physical foundaDons of the superconducDng phenomenon in order to suggest with certainty, the line of experimental process to obtain superconducDvity at room temperature in viable condiDons for its applicaDon to large-scale. An example of the predicDve effect of the theoreDcal approach on superconducDvity was observed in the idea proposed by Ashcrom (cita), who stated that hydrogen-rich systems would be viable candidates to be high criDcal temperature (Tc) superconductors. This proposal gave rise to experimentaDon in this field with the discovery of new high-Tc superconductors, as H3S (Tc of 203 K at 155 GPa) or LaH10 (Tc of 260 K at 180 GPa), called hydride superconductors. This discovery gave a new impetus to this field of study, which had been stuck with the superconducDng cuprates (Tc of 150 K) since 1995. The current difficulty with the hydride superconductors is in their high-pressure condiDons of formaDon.
In this sense, evaluaDng possible new ways to predict Tc values in terms of well-defined physical condiDons (such as pressure, doping, etc.) is an interesDng line of work. Here, the study of the derivaDve seeks to establish the possible existence of pa_erns that lead to the determinaDon of an opDmum temperature of the system (superconducDng criDcal temperature), which would also avoid the use of test or experimental Tc in first-principles calculaDons." As can be seen in the manuscript: (2) The authors menDon in the abstract that their findings support the hypothesis that H3S under 155 GPa pressure achieves the highest experimental Tc. However, this is not discussed in detail in the main text. It would be helpful to explain how the results in Figs The pa_erns of the derivaDve vs. Tc in the H3S reveal that these seem to have a characterisDc behavior at a specific pressure (155 GPa).
As can be seen in the manuscript: (3) The authors menDon that μ* shows an almost linear correlaDon with temperature in Fig. 4, but it is unclear from the figure whether this is true for all pressures or only for certain ones. It would be helpful to clarify this point in the capDon or main text.

Answer:
The wording of the sentence was improved to make the idea clearer.
It is observed in Fig.4 that μ* vs T has a comparable trend between pressures. This could be assumed to be quasi-linear in a first approximaDon. However, this quasi-linearity is much more evident at higher pressure.

As can be seen in the manuscript:
The final version of the manuscript is a_ached (LaTeX format), in which all the adjustments made have been highlighted.
We hope that we have been clear in each answer.
Thank you for your consideraDon of our work,